On Whitney-type extension theorems for $C^{1,+}$, $C^2$, $C^{2,+}$, and $C^3$-smooth mappings between Banach spaces
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866909686828105728 |
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| author | Johanis, Michal |
| author_facet | Johanis, Michal |
| contents | In 1973 J. C. Wells showed that a variant of the Whitney extension theorem holds for $C^{1,1}$-smooth real-valued functions on Hilbert spaces. In 2021 D. Azagra and C. Mudarra generalised this result to $C^{1,ω}$-smooth functions on certain super-reflexive spaces. We show that while the vector-valued version of these results do hold in some rare cases (when the target space is an injective Banach space, e.g. $\ell_\infty$), it does not hold for mappings from infinite-dimensional spaces into "somewhat euclidean" spaces (e.g. infinite-dimensional spaces of a non-trivial type), and neither does the $C^2$-smooth variant. Further, we prove negative results concerning the real-valued $C^{2,+}$, $C^{2,ω}$, and $C^3$-smooth versions generalising older results of J. C. Wells. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Whitney-type extension theorems for $C^{1,+}$, $C^2$, $C^{2,+}$, and $C^3$-smooth mappings between Banach spaces Johanis, Michal Functional Analysis 46G05, 46T20 In 1973 J. C. Wells showed that a variant of the Whitney extension theorem holds for $C^{1,1}$-smooth real-valued functions on Hilbert spaces. In 2021 D. Azagra and C. Mudarra generalised this result to $C^{1,ω}$-smooth functions on certain super-reflexive spaces. We show that while the vector-valued version of these results do hold in some rare cases (when the target space is an injective Banach space, e.g. $\ell_\infty$), it does not hold for mappings from infinite-dimensional spaces into "somewhat euclidean" spaces (e.g. infinite-dimensional spaces of a non-trivial type), and neither does the $C^2$-smooth variant. Further, we prove negative results concerning the real-valued $C^{2,+}$, $C^{2,ω}$, and $C^3$-smooth versions generalising older results of J. C. Wells. |
| title | On Whitney-type extension theorems for $C^{1,+}$, $C^2$, $C^{2,+}$, and $C^3$-smooth mappings between Banach spaces |
| topic | Functional Analysis 46G05, 46T20 |
| url | https://arxiv.org/abs/2507.09384 |